Secondary Interaction Contribution in Hydrogen-Bonded Complex:
Theoretical Model Study in Hydrogen Fluoride Trimer

Shun-ichi KAWAHARA and Tadafumi UCHIMARU

1 Introduction

Figure 1. Watson-Crick type base pair and hydrogen bond arrangement in a plurally hydrogen-bonded complex.

According to Jorgensen's proposal, the total hydrogen bond should be more stable when all the proton donor-acceptor pairs are arranged in the same direction as in Figure 1 I: DD-AA type complex, for example, the arrangement of the hydrogen bonds B and C in a guanine - cytosine base pair. On the other hand, the total hydrogen bond should be less stable when the proton donor-acceptor pairs are arranged alternately as in Figure 1 II: DA-AD type complex, for example, the arrangement of the hydrogen bonds A and B in guanine - cytosine and adenine - uracil base pairs. As a result, the adenine - uracil base pair is considered to have two hydrogen bonds and two repulsive secondary interactions (A-A and D-D type), and the guanine - cytosine base pair is considered to have three hydrogen bonds, two attractive secondary interactions (D-A type), and two repulsive secondary interactions (A-A and D-D type) [6]. We already reported the importance of secondary electrostatic interactions in hydrogen-bonding complexes [7], and we report herein the contribution of the secondary interactions to the whole hydrogen bond formation energy, using hydrogen fluoride dimer/trimer as the models of the hydrogen bond site of a plurally hydrogen-bonded complex.

2 Computational Methods

Figure 2. Modeling for divided hydrogen bond sites of plurally hydrogen-bonded complex by the arrangement of the hydrogen fluoride dimers/trimers.

Figure 3. Estimation of the hydrogen bond (x), the repulsion of neighboring HF (y) and the secondary interaction (z) for the model of neighboring hydrogen bond sites by the combination of hydrogen fluoride trimer division (l, m and n).

Figure 2 V-VII show the arrangement of three hydrogen fluorides for the D-A type attractive secondary interaction (Figure 2 V), the D-D type repulsive secondary interaction (Figure 2 VI) and the A-A type repulsive secondary interaction (Figure 2 VII). The magnitude of each interaction was evaluated as follows. The hydrogen fluorides were named as shown in Figure 3. D-A type arrangement of three hydrogen fluorides (Figure 2 V) is shown for example. Three molecular interaction energies should be considered in the system: x = hydrogen bond interaction between HF^A and HF^B (EH_B), y = repulsive interaction between HF^A and HF^C (E_rep), and z = secondary interaction between HF^B and HF^C (E₂). However, these interactions could not be calculated directly by the supermolecular method; thus, to determine these three interactions, three interaction energies, i. e., l: total stabilization energy of (HF^A-HF^C)+HF^B (interaction energy between (HF^A-HF^C) and HF^B, here, HF^A-HF^C were dealt as one part. See also Figure 3), m: interaction energy of (HF^B-HF^C)+ HF^A and n: interaction energy of (HF^A-HF^B)+HF^C were calculated. The interaction l is considered to include the interactions x and z, the interaction m is considered to include the interactions x and y, and the interaction n is considered to include the interactions y and z; thus, l = x + z, m = x + y, and n = y + z. D-D and A-A type secondary interactions were also calculated as above, using the corresponding trimer complex (Figure 2 VI and VII, respectively)
All molecular orbital calculations were carried out using a Gaussian 94 program [11].

3 Results and Discussion

Table 1. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride dimer models, in B3LYP/6-311+G**.

	Energy	Ratio	Model in Figure 2
EHB	-3.82	-	I
E2 D-A	-1.07	0.28	II
E2 D-D	-0.14	0.04	III
E2 A-A	+0.77	0.20	IV

In the trimer systems, three-body interaction should be considered [12]; however, three-body interactions in the D-A type, the D-D type and the A-A type trimer (E_T = E_total - (E_HB + E₂ + E_rep)) were all small, comparing total interaction energies of the complexes (E_total = E_HFtrimer - 3E_HFmonomer). Rincon et al. reported hydrogen bond cooperativity in the hydrogen fluoride clusters. The stabilization energy per one hydrogen bond in the cluster significantly arises from the tetramer, but it is not remarkably changed between the dimer and the trimer [13]. Our results were in good agreement with the results of Rincon et al. [13]; thus, we can safely deal with these trimer systems as the model of hydrogen bond sites without consideration of three-body interaction. Taking account of these results, MP2and QCISD(T) studies were carried out based on the hydrogen fluoride dimer systems. Table 3 shows the hydrogen bond energy and three types of secondary interaction energies in various methods using the hydrogen fluoride dimer systems. The basis set effect in the MP2 method was small. The D-A type secondary interaction was estimated to be about 30%. The result of D-A type attractive secondary interaction was similar to that of DFT. The D-D type secondary interaction was evaluated to be about 10% (QCISD(T)) or about 15% (MP2) attractive. The A-A type repulsive secondary interaction was about 10-15% of the hydrogen bond. Thus, DFT estimated D-A type secondary interaction in high accuracy, but it underestimated the contribution of the D-D type and overestimated the A-A type secondary interaction.

Table 2. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride trimer models, in B3LYP/6-311+G**.

	Interaction Energy			Interaction in Figure 3	Ratio(z/x)
EHB	-3.52	-3.85	-3.85	x	-
Erep+	+11.64	+5.13	+5.13	y
E2 D-A	-0.79			z	0.22
E2 D-D		-0.19		z	0.05
E2 A-A			+0.72	z	0.19
ET	-0.29	+0.05	+0.04	three-body interactions
Etotal	+7.04	+1.13	+2.04	total interaction
	V	VI	VII	Model in Figure 2

+Repulsive interaction between HF^A and HF^C in Figure 3. See the computational method in the text.

Figure 4. Potential energy curves of the A-A type (cross), DD type (circle), D-A type (plus) secondary interactions, and hydrogen bond (square) depending on the hydrogen bond length (between the heavy atoms). See also Figure 2.

Figure 4 shows the change in the secondary interaction depending on the distance between the hydrogen fluorides, which corresponds to the hydrogen bond length, in DFT (A) and QCISD(T) (B) methods. The trend of these two figures is similar: all interactions in DFT are a parallel shift of the results in QCISD(T). The D-A type and the A-A type interactions were quite insensitive to the hydrogen bond length. In contrast to these two interactions, the D-D type interactions were sensitive to the hydrogen bond length. It was repulsive when the distance between the hydrogen fluorides was long, and it was attractive when the distance between the hydrogen fluorides was short. As the result, it was weakly attractive in the range of normal hydrogen bond length. From the result of this potential energy study, Figure 5 is considered to be the reason why the D-D type secondary interaction was evaluated as attractive: there are two tertiary attractive interactions between the proton and the fluorine atom of the opposite side, and these two interactions overcome the repulsive interaction between two protons (Figure 5A) [14]. In the case of the D-D interaction, tertiary interaction is 52 degrees bent hydrogen bond-like interaction. The atom distance between the hydrogen and the fluorine (R_HF = ca. 2.2A) was comparable with the atom distance in the hydrogen bond (1.8 A). On the other hand, the R_HF in A-A type arrangement (Figure 5B) was much longer (ca. 4.0 A). The fluorine-hydrogen-fluorine bent angle (ca. 149 degrees ) is too highly bent to form the hydrogen bond-like interaction. Thus, tertiary interaction in A-A type arrangement is considered to be negligible.

Table 3. Molecular interaction (the hydrogen bond and the secondary interaction) energies (kcal/mol) and the ratio of the secondary interaction based on the hydrogen bond stability by use of hydrogen fluoride dimer models, in various calculation levels.

	B3LYP/6-311+G**		MP2/aug-cc-pVTZ		MP2/basis set limit+		QCISD(T)/aug-cc-pVTZ
	Energy	Ratio	Energy	Ratio	Energy	Ratio	Energy	Ratio
H-Bond	-3.82	-	-3.24	-	-3.59	-	-3.35	-
2nd D-A	-1.07	-0.28	-1.08	-0.33	-1.08	-0.31	-1.07	-0.32
2nd D-D	-0.14	-0.04	-0.48	-0.15	-0.51	-0.14	-0.30	-0.09
2nd A-A	+0.77	+0.20	+0.47	+0.15	+0.43	+0.12	+0.44	+0.13

+: The values calculated based on the basis set limit, estimated by Feller's procedure [10].

From the results in Figure 4, both D-A type and A-A type secondary interaction energies were almost independent of the hydrogen bond length. In contrast to these two types of secondary interaction energies, D-D type secondary interaction energies were highly dependent on the hydrogen bond length. It was repulsive in the case of longer hydrogen bond length, and it was attractive in the case of shorter hydrogen bond length. Thus, the hydrogen bond length of the DA-AD type complex should be shorter than that of the DD-AA type, if the hydrogen bond capability of each hydrogen bond sites is the same. On the other hand, total hydrogen bond stabilization in the DD-AA type complex should be larger than that of the DA-AD type complex.

Figure 5. The tertiary interaction of D-D/A-A arrangement.

Next, a model study using a formamide-hydrogen fluoride complex was attempted. There are two proton donor (D1 and D2 in Figure 6) sites and acceptor (A1 and A2) sites in formamide. The hydrogen bond sites A1 and A2 are proton acceptor sites without and with D-D type secondary interaction. The hydrogen bond sites D1 and D2 are proton donor sites without and with A-A type secondary interaction. The proton donatability of D1 and D2 sites, and acceptability of A1 and A2 sites without the secondary interaction should be similar, respectively. Thus, the contribution of the secondary interaction should be observed as the difference in the hydrogen bond stability.

Figure 6. Hydrogen bond sites of formamide.

Table 4 shows the results in B3LYP/6-311+G** and MP2/aug-cc-pVTZ levels of the calculation. In both calculation levels, change in the stabilization energy with and without the secondary interaction was in good agreement with the results from the model complex using hydrogen fluorides (compare the ratio of the secondary interactions based on the hydrogen bond energy of Tables 3, 4). D-D type secondary interaction was a very small attractive force; on the other hand, A-A type secondary interaction was a large repulsive force, and the ratio of the A-A type secondary interaction in the total interaction was 0.2-0.3. Thus, the secondary interaction contribution in a plurally hydrogen-bonded complex, shown in Table 3, is applicable for rough estimation of the trend in stabilization of the plurally hydrogen-bonded complex.

Table 4. Total interaction and the secondary interaction energy (kcal/mol) and the ratio of the secondary interaction in formamide-HF complex.

	B3LYP/6-311+G**		MP2/aug-cc-pVTZ
H-Bond Site	Total Int.	2nd Int. (ratio/type)	Total Int.	2nd Int. (ratio/type)
D1	-2.37		-2.30
D2	-1.70	0.67 ( 0.28 /A-A)	-1.76	0.54 ( 0.24 /A-A)
A1	-9.76		-9.68
A2	-9.73	0.03 ( 0.003 /D-D)	-9.33	0.35 ( 0.04 /D-D)

4 Conclusion

The services and computational time made available by the National Institute of Advanced Industrial Science and Technology (AIST) have been essential to this study and are gratefully acknowledged.

References